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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubclsubN | Structured version Visualization version GIF version |
Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclsub.s | β’ π = (PSubSpβπΎ) |
psubclsub.c | β’ πΆ = (PSubClβπΎ) |
Ref | Expression |
---|---|
psubclsubN | β’ ((πΎ β HL β§ π β πΆ) β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 β’ (β₯πβπΎ) = (β₯πβπΎ) | |
2 | psubclsub.c | . . 3 β’ πΆ = (PSubClβπΎ) | |
3 | 1, 2 | psubcli2N 39467 | . 2 β’ ((πΎ β HL β§ π β πΆ) β ((β₯πβπΎ)β((β₯πβπΎ)βπ)) = π) |
4 | eqid 2725 | . . . . . . 7 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
5 | 4, 1, 2 | psubcliN 39466 | . . . . . 6 β’ ((πΎ β HL β§ π β πΆ) β (π β (AtomsβπΎ) β§ ((β₯πβπΎ)β((β₯πβπΎ)βπ)) = π)) |
6 | 5 | simpld 493 | . . . . 5 β’ ((πΎ β HL β§ π β πΆ) β π β (AtomsβπΎ)) |
7 | psubclsub.s | . . . . . 6 β’ π = (PSubSpβπΎ) | |
8 | 4, 7, 1 | polsubN 39435 | . . . . 5 β’ ((πΎ β HL β§ π β (AtomsβπΎ)) β ((β₯πβπΎ)βπ) β π) |
9 | 6, 8 | syldan 589 | . . . 4 β’ ((πΎ β HL β§ π β πΆ) β ((β₯πβπΎ)βπ) β π) |
10 | 4, 7 | psubssat 39282 | . . . 4 β’ ((πΎ β HL β§ ((β₯πβπΎ)βπ) β π) β ((β₯πβπΎ)βπ) β (AtomsβπΎ)) |
11 | 9, 10 | syldan 589 | . . 3 β’ ((πΎ β HL β§ π β πΆ) β ((β₯πβπΎ)βπ) β (AtomsβπΎ)) |
12 | 4, 7, 1 | polsubN 39435 | . . 3 β’ ((πΎ β HL β§ ((β₯πβπΎ)βπ) β (AtomsβπΎ)) β ((β₯πβπΎ)β((β₯πβπΎ)βπ)) β π) |
13 | 11, 12 | syldan 589 | . 2 β’ ((πΎ β HL β§ π β πΆ) β ((β₯πβπΎ)β((β₯πβπΎ)βπ)) β π) |
14 | 3, 13 | eqeltrrd 2826 | 1 β’ ((πΎ β HL β§ π β πΆ) β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3940 βcfv 6542 Atomscatm 38790 HLchlt 38877 PSubSpcpsubsp 39024 β₯πcpolN 39430 PSubClcpscN 39462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-proset 18284 df-poset 18302 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p1 18415 df-lat 18421 df-clat 18488 df-oposet 38703 df-ol 38705 df-oml 38706 df-ats 38794 df-atl 38825 df-cvlat 38849 df-hlat 38878 df-psubsp 39031 df-pmap 39032 df-polarityN 39431 df-psubclN 39463 |
This theorem is referenced by: pclfinclN 39478 |
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